What is the angular speed of earth?
\[\begin{align}
  & \left( a \right)\dfrac{\pi }{30}\dfrac{rad}{s} \\
 & \left( b \right)\dfrac{\pi }{1800}\dfrac{rad}{s} \\
 & \left( c \right)\dfrac{\pi }{21600}\dfrac{rad}{s} \\
 & \left( d \right)\dfrac{\pi }{43200}\dfrac{rad}{s} \\
\end{align}\]

Hint: Angular speed is how quickly an object rotates. It is defined as the change in the angle of the object per unit time .Radian is the way of measuring angles. Here, by using angular velocity formula we can find angular speed of the earth which is denoted by w. Its SI unit is radian per second.

Complete step-by-step solution:
Angular velocity is the rate of change of the position angle of the object with respect to time. Angular velocity is also known as angular frequency vector. It refers to how fast an object rotates or revolves when it is in a relative position to another point. There are two types of angular velocity which are orbital angular velocity and spin angular velocity. Above or below the plane is the direction of velocity. We can find the direction of angular velocity using the thumb rule, that is if we hold the axis with our right hand and rotate our fingers in the direction of motion of the body then the thumb will point the direction of angular velocity. If any object rotates about an axis, angular velocity is the same at every point.
Angular velocity w is associated with the rotational motion of the particle p. The quantity is always measured in \[\dfrac{rad}{s}\]. Angular acceleration is alpha which is associated with rotational motion of the particle. The angular velocity is a pseudo vector. Pseudo vector is one whose direction is perpendicular to the plane of rotational movement. Greater is the rotational angle we get greater angular velocity which is a vector quantity.
Angular velocity is the rate of change of angular displacement and can be described by the relationship between them such as:
\[w=\dfrac{\theta }{t}\]
Here, w= angular velocity, theta = position angle and t = time.
 We know that, 2 pi is equal to 360 degree or full circle so,\[\theta =2\pi \], Using the value of theta in above formula we get:
\[w=\dfrac{2\pi }{24\times 60\times 60}\]
On simplifying above equation we get:
\[w=\dfrac{2\pi }{86400}\]
On solving above equation we get our answer which is:
\[w=\dfrac{\pi }{43200}\dfrac{rad}{s}\]
 Therefore, the angular speed of the earth is \[w=\dfrac{\pi }{43200}\dfrac{rad}{s}\].
Hence, the correct option is d.

Note:Angular velocity or speed was first calculated by Leonhard Euler. The gravitational force causes the change in the angular velocity. Tachometer is the device for indicating the rotator speed of a rotating object .It shows engine speed in thousands of revolutions per minute.